Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)? - MathOverflow

Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?

2010-11-29T18:20:48Z

In his 1967 paper A convenient category of topological spaces, Norman Steenrod introduced the category CGH of compactly generated Hausdorff spaces as a good replacement of the catgory Top topological spaces, in order to do homotopy theory.

The most important defference between CGH and Top is that in CGH there is a functorial homeomorphism $$\mathrm{map}(X,\mathrm{map}(Y,Z))\cong \mathrm{map}(X\times Y,Z),$$ a fact that is only true in Top under the extra assumption that $Y$ is locally compact.

But in more recent papers, I see that people use CGWH spaces instead of CGH spaces... Why?

Could someone explain to me what goes wrong in CGH spaces (please illustrate with an example),
and explain how the "w" fixes everything?

Also (following Jeff's comment), to whom should the "w" be attributed?

One more wish: can someone give me an example of a CGWH space that isn't CGH?

Answer by Harry Gindi for Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?

2010-11-29T21:19:24Z

Are you sure you didn't mean CG-spaces (without the Hausdorff assumption)? CGH isn't even closed under colimits (it's pretty easy to construct the real line with two origins), so they don't seem to have all of the nice categorical properties we would want.

I think that the real question is why we need weak Hausdorffness instead of just compact generation.

The above is incorrect (although it seems that Dan Ramras and I have made the same mistake, as well as the paper cited by Andrey), but here's a paper by Neil Strickland that you might find interesting: Click!.

Answer by Andrey Rekalo for Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?

2010-11-29T21:21:36Z

A web search suggests that the category of CGWH spaces was introduced in the paper "Classifying Spaces and Infinite Symmetric Products" by M. C. McCord (Transactions of the American Mathematical Society Vol. 146, (1969), pp. 273-298).

McCord motivated introduction of his "weak Hausdorff" separation axiom by noting that

"the requirement of the Hausdorff condition can be a problem because certain standard operations on spaces can lead outside the category", in particular quotient spaces in algebraic topology and topological algebra.

Answer by Charles Rezk for Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?

2010-11-29T21:27:18Z

I believe that CGWH spaces were first used in a systematic way in the work of Lewis-May-Steinberger on spectra. It is certainly the case that Gaunce Lewis's (unpublished) thesis contains the best reference on CGWH spaces that I'm aware of. (I haven't looked at the McCord paper Andrey mentions. Update: Having looked at McCord's paper, it does indeed seem to be the one to introduce CGWH (the idea of which he attributes to J.C. Moore.))

As to why one might prefer to use CGWH spaces, I'm not precisely sure. But here is one possibility.

A key property of the category of CG spaces is that the product of a quotient map with a space is still a quotient map. In CGWH spaces, something even nicer is true: any pullback of a quotient map (along any map) is still a quotient map. (I don't know whether this nicer fact fails in CGH, but I suspect it does.)

Another nice fact about CGWH: regular monomorphisms are precisely the closed inclusions.("Regular monomorphism" means the monomorphism is an equalizer of some pair.) (I originally said here that regular epis in CGWH are precisely quotient maps, but on reflection I'm not sure this is true.)

Answer by Dan Ramras for Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?

2010-11-29T21:53:53Z

To flesh out my comment above: in the Errata to Geometry of Iterated Loop Spaces (p. 485 here: http://www.math.uchicago.edu/~may/BOOKS/homo_iter.pdf) May states that he should have used weak Hausdorff spaces "in order to validate some limit arguments." I'm not sure exactly what he means; in particular I would think he really means colimit arguments.